Potential Energy Formula: Equations and Solved Examples

Potential energy is the energy stored in an object due to its position. It can be classified as Gravitational Potential Energy, Electric Potential Energy, etc. Potential Energy Formula depends on the force acting on the two objects.

• Gravitational potential energy is the energy of an object due to its height above the ground.
• Elastic potential energy is stored in a deformed object, such as a stretched rubber band.
• The electric potential energy of an electric charge is the energy that an object has due to its position in an electric field.
• Potential energy is responsible for the motion of planets, springs and pendulums, and the behaviour of electric circuits.

Key Terms: Potential Energy, Gravitational Potential Energy, Elastic Potential Energy, Energy, Force, Electric Charge

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Potential Energy Formula

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The potential energy formula depends on the type of potential energy. The most common types of potential energy include:

Gravitational Potential Energy

The formula for gravitational potential energy is – 

U_g = mgh

Where

• U_g = Potential Energy of a given object
• m = mass of the object
• g = acceleration due to gravity
• h = height of the object

Example – A ball held above the ground has gravitational potential energy due to its position in the gravitational field of the Earth. 

Elastic Potential Energy

As elasticity is dependent on the force exerted by the object force, in this case, is defined as

U_e = ½ kx²

Where, 

• U_e​ is the elastic potential energy
• k is the spring constant 
• x is the displacement of the spring from its equilibrium position 

Example – Elastic potential energy is stored in a spring when it is stretched. When it is released, the elastic potential energy is converted into kinetic energy.

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Electric potential energy

It is the energy required to bring a charged particle from infinity to a given point in an electric field.

U_E ​= k_e​\(q_1q_2 \over r\)

Where

• U_E is the electric potential energy 
• k_e​ is Coulomb's constant (9×10⁹ Nm²/C²)
• q1​ and q2​ are the charges of the two objects in coulombs
• r is the distance between the two objects in meters

Example – When a battery is not connected to a circuit, the electric potential energy is stored in the battery. 

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Derivation of Potential Energy Formula

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The potential Energy formula can be derived from the following equation –

W = Fd

• where W is the work done,
• F is the force applied
• d is the displacement.

We can derive the equation by considering a conservative force. Considering a situation where an object of mass m is lifted from a height h₁ to h₂, the force required to lift the object can be expressed as a product of mass into gravitational distance = mg

So, F = mg

The displacement of the object is equal to the difference in height between the two positions, which is given by d = h₂ – h₁. Hence, work done to lift the object is \(W = mg(h_2 - h_1)\). 

Since potential energy is the ability of an object to do work, the potential energy of the object at height h₂ is given by 

\(U = mg(h_2 - h_1)\)

This is the potential energy formula for gravitational potential energy.

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Solved Examples

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Example 1: Calculate the gravitational potential energy of an object with a mass of 5 kg when lifted to a height of 10 meters above the ground. (Use g = 9.8 m/s²)

Solution: The formula for gravitational potential energy (PE) is given by:

U = mgh

where:
- m is the mass of the object,
- g is the acceleration due to gravity (9.8 m/s²),
- h is the height above the reference point.

Substitute the given values into the formula:

Potential Energy = 5 x 9.8 x 10

= 490 Joules

Therefore, the gravitational potential energy of the object is 490 Joules.

Example 2: A spring has a potential energy of 100 Joules when compressed. If the spring constant is (k = 50 N/m), find the compression distance.

Solution: The formula for the potential energy stored in a spring is given by: \([ PE = \frac{1}{2} k x^2 ]\)

where:
- k is the spring constant,
- x is the compression or stretching distance.

Rearrange the formula to solve for x:

\([ x = \sqrt{\frac{2 \cdot PE}{k}} ]\)

Substitute the given values into the formula:

\([ x = \sqrt{\frac{2 \cdot 100 \, \text{J}}{50 \, \text{N/m}}}]\)

\([ x = \sqrt{4} ]\)

x = 2m

Therefore, the compression distance of the spring is 2 meters.

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Things to Remember

• Potential energy is the energy an object possesses due to its position or condition.
• Gravitational Potential Energy is associated with an object's position in a gravitational field.
• Elastic Potential Energy is stored in objects undergoing deformation, like springs.
• Gravitational Potential Energy Formula = m⋅g⋅h, where m is mass, g is acceleration due to gravity, and h is height.
• Elastic Potential Energy Formula = ½ ​kx², where k is the spring constant, and x is the compression or stretching distance.
• According to the law of conservation of energy, the total mechanical energy (kinetic + potential) of an isolated system remains constant.
• Potential energy is measured in joules (J) in the International System of Units (SI).